Generalized factorization in ℤ∕mℤ

Mahlum, Austin; Mooney, Christopher Park

doi:10.2140/involve.2016.9.379

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Generalized factorization in $\mathbb{Z}/m\mathbb{Z}$

Austin Mahlum and Christopher Park Mooney

Vol. 9 (2016), No. 3, 379–393

DOI: 10.2140/involve.2016.9.379

Abstract

Generalized factorization theory for integral domains was initiated by D. D. Anderson and A. Frazier in 2011 and has received considerable attention in recent years. There has been significant progress made in studying the relation $τ_{n}$ for the integers in previous undergraduate and graduate research projects. In 2013, the second author extended the general theory of factorization to commutative rings with zero-divisors. In this paper, we consider the same relation $τ_{n}$ over the modular integers, $ℤ ∕ m ℤ$ . We are particularly interested in which choices of $m, n \in ℕ$ yield a ring which satisfies the various $τ_{n}$ -atomicity properties. In certain circumstances, we are able to say more about these $τ_{n}$ -finite factorization properties of $ℤ ∕ m ℤ$ .

Keywords

modular integers, generalized factorization, zero-divisors, commutative rings

Mathematical Subject Classification 2010

Primary: 13A05, 13E99, 13F15

Milestones

Received: 27 September 2014

Revised: 7 April 2015

Accepted: 6 June 2015

Published: 3 June 2016

Communicated by Vadim Ponomarenko

Authors

Austin Mahlum
Department of Mathematics Viterbo University La Crosse, WI 54601 United States

Christopher Park Mooney
Department of Mathematics Westminster College Fulton, MO 65251 United States

Vol. 9, No. 3, 2016